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9.4. Extended Euclid’s Algorithm
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Hello, class! Today we are going to discuss Bezout’s theorem, which tells us that the GCD of two integers can be expressed as a linear combination of those integers. Can anyone define what GCD means?
GCD stands for greatest common divisor, right? It's the largest number that divides both integers.
Exactly right! For example, if we have the numbers 6 and 14, their GCD is 2. So, can anyone help me express that using Bezout's theorem?
It would be 2 = 1*6 + (-2)*14, where s is 1 and t is -2.
Good job! This shows how we can represent the GCD as a linear combination. Remember, the integers s and t can be positive or negative, and that’s crucial for our discussions ahead.
So, can we always find such integers s and t for any pair of numbers, or is there a limitation?
That’s a great question! Yes, for any two integers a and b, there will always exist integers s and t such that GCD(a, b) = sa + tb. That's the essence of Bezout's theorem!
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Now that we understand Bezout’s theorem, let’s talk about the Extended Euclidean Algorithm. This is an algorithm that allows us to find the GCD of two integers and also calculates the Bezout's coefficients.
What makes it 'extended'? What is different from the regular Euclidean algorithm?
Great question! The regular Euclidean algorithm finds the GCD, but the extended version keeps track of additional information so that we can also find s and t. Can anyone think of why that might be useful?
It could help in modular arithmetic, especially when finding inverses!
Exactly! Let's say we are trying to solve for a modular inverse of a number; we'd need those coefficients to express it in a usable form.
So how would we go about finding those s and t using this algorithm?
We’ll apply the algorithm iteratively while expressing the remainders we get as linear combinations of a and b. We'll go over a specific example shortly.
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Let’s illustrate this algorithm with an example. Suppose a is 252 and b is 198, can anyone outline how we would start?
We would apply the regular Euclidean algorithm first to find the GCD.
Exactly! After performing the Euclidean steps, how do we write these steps down to get back to s and t?
We express each remainder in terms of a and b at each step!
Spot on! By substituting back, we can eventually find our Bezout’s coefficients. Does anyone want to try substituting and verifying?
Sure! We’d find that 18 can be represented as 4*252 + (-5)*198. So here, s is 4 and t is -5.
Well done! That proves 18 can be formed by a linear combination, fulfilling Bezout's theorem.
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Now, let's connect everything to multiplicative inverses modulo N. When do we say an integer a has a multiplicative inverse?
If the GCD of a and N is 1, then a has a multiplicative inverse.
Absolutely! So, using our earlier example, if 198 has to be the modulus, would 252 have an inverse?
Yes, because GCD(252, 198) is 18, not 1. So, 252 won’t have an inverse in mod 198.
Correct! And remember that finding the inverse using the coefficients from the Extended Euclidean Algorithm can save us a lot of time!
So, how do we actually compute and find that inverse?
We apply the Extended Euclidean Algorithm until we find that suitable s such that s*a mod N = 1, which can then be used to express the multiplicative inverse.
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To summarize today's lesson: we learned about Bezout’s theorem, the Extended Euclidean Algorithm, and their applications in finding GCD and multiplicative inverses. Can anyone recap what Bezout’s theorem states?
It states that any GCD can be expressed as a linear combination of two integers.
Right! And why do we need the Extended version?
To also find those coefficients s and t, which helps in many applications.
Exactly! Remember, whenever you're working with modular arithmetic, being able to find that multiplicative inverse is crucial.
Thanks, this really helps clarify how these concepts connect!
I'm glad to hear that! Be sure to practice these concepts further in your exercises.
Overview
Short Summary
The Extended Euclidean Algorithm provides a method to find the greatest common divisor (GCD) of two integers and expresses it as a linear combination of the two integers.
Medium Summary
This section explores Bezout’s theorem, which states that the GCD of two integers can be represented as a linear combination of those integers. It introduces the Extended Euclidean Algorithm as a means to find the GCD and the coefficients in that linear combination, providing practical implications in computing multiplicative inverses modulo N.
Detailed Summary
In this section, we delve into the Extended Euclidean Algorithm, which builds upon Bezout’s theorem. \nFirst, we establish that for any two integers a and b, there exist integers s and t such that GCD(a, b) = sa + tb, where s and t are Bezout's coefficients. This theorem is fundamental in number theory and has practical applications in algorithm design, particularly in cryptography. \nThe proof comprises several claims about the properties of the set S of integer linear combinations of a and b, culminating in the demonstration that GCD(a, b) can indeed be expressed in this form. \nFurthermore, we explore how this process can be streamlined using the Extended Euclidean Algorithm, which not only computes the GCD but also finds the necessary coefficients s and t through thoughtful bookkeeping during the algorithm's execution. This approach significantly enhances our ability to solve equations in modular arithmetic and is pivotal for finding multiplicative inverses when we know that a and b are coprime.
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Create a free accountSo, the goal is to find those integer linear combiners. These integer linear combiners, they are called as Bezout’s coefficient. So, you might be wondering why at the first place, I am interested to find them; later on when we will discuss about modular multiplicative inverse, this Bezout’s coefficients will come very handy, so that is why we want to find them. It turns out that by doing some extra bookkeeping, that means by maintaining some additional values and data structure in my Euclid algorithm, which are used for finding out the GCD of a and b, I can find out the Bezout’s coefficient as well.
Detailed Explanation
Extended Euclid’s Algorithm not only helps us find the GCD of two integers but also allows us to identify the integers (s and t) that can create this GCD through a linear combination of the two numbers. Understanding these coefficients (Bezout’s coefficients) is crucial for later topics, such as finding modular inverses. This algorithm adds extra steps, known as bookkeeping, to track values that will ultimately yield the coefficients.
Examples & Analogies
Imagine you're trying to balance a budget with various expenses and incomes. You not only need to know if you have enough to cover your costs (like finding the GCD), but you also want to know how much of each source contributes to that balance (like finding the Bezout’s coefficients). The Extended Euclid’s Algorithm helps you keep track of those contributions while ensuring that you remain balanced.
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Create a free accountNow, let us see how exactly the various remainders are computed during the execution of the Euclid’s GCD algorithm. So, in my first step, this will be my x (252) and this will be my y (198). And this will be my r (54). In the next iteration, this will become my x (198), my current r will become next y (54) and this will be the next r (36). So, the underlined things are the remainders that I am obtaining.
Detailed Explanation
The Extended Euclid's Algorithm starts similarly to the basic Euclidean algorithm. For instance, if we wish to find the GCD of 252 and 198, we initiate the process with these two numbers. We repeatedly calculate the remainders and adjust our variables according to the algorithm's rules. During these calculations, we keep track of how each remainder relates back to the original numbers, setting the stage to eventually express the GCD as a linear combination of the two inputs.
Examples & Analogies
Consider a water jug problem where you need to measure precisely a certain amount of liquid using two different jugs. Each time you pour water from one jug to another, you note how much is poured and calculate how much is left (the remainder). This process of tracking how much liquid you have left mirrors the way we track remainders in the Extended Euclid's Algorithm.
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Create a free accountAnd then I get my Bezout’s coefficients as 4 and -5. So, basically what we have to do is at each step, we have to keep track of my quotients, and the remainder and quotients and the remainder, that is why I have underlined them. So, in this demonstration, I have actually done a backward pass, because we went all the way back and stopped where 54 was expressed in terms of 252 and 198.
Detailed Explanation
After computing the GCD, we need to express that GCD as a linear combination of the original integers. Through back substitution, we replace each remainder with its expression in terms of the original numbers until we reach the final form. This process reveals the Bezout's coefficients, allowing us to solve for s and t, confirming that the GCD can indeed be expressed in this way.
Examples & Analogies
Imagine building a bridge where each section is determined by a previous section length. Each time you complete a section, you note how it connects back to the foundation. This concept of referring back to previous structures is like back substitution in the algorithm, where we link each remainder back to the original integers until we form the complete resultant structure.
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Key concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
- Bezout's theorem:
The GCD of two integers can be expressed as a linear combination of those integers.
- Extended Euclidean Algorithm:
An algorithm that computes the GCD and the Bezout coefficients.
- Multiplicative Inverse:
The process of using GCD and Bezout's theorem to compute inverses in modular arithmetic.
Examples
Memory aids
Imagine GCD is a treasure, buried between two hills a and b. S and t are the brave adventurers who find the way to express it using their unique skills.
To remember the steps of the Extended Euclidean Algorithm: 'Find, Divide, Express.' Clearly defines the operations we do at each step.
Flash Cards
Glossary
GCD
Greatest Common Divisor; the largest integer that divides two numbers without leaving a remainder.
Bezout's Theorem
A theorem stating that the GCD of two integers can be expressed as a linear combination of those integers.
Extended Euclidean Algorithm
An algorithm that computes the GCD of two integers and the coefficients of the linear combination used in Bezout's theorem.
Multiplicative Inverse
An integer b is the multiplicative inverse of a modulo N if (a * b) mod N = 1.
Co-prime
Two integers are co-prime if their GCD is 1.